The Beauty of Fractals

JS Communications Skills – Task 7: Physics Blog Entry

by Maxim Mulligan

The Fractals All Around Us

What are fractals? I’m sure many of you have seen them before in some context, perhaps unaware of what they actually are, perhaps just captivated by their beauty. Fractals are something which commonly occur in nature, some of you readers might be familiar with the self similar nature of foam from soap. Self similar is a term you will hear quite a lot when fractals are being discussed so it would be good to familiarize ourselves with its meaning. Self similar objects have either exact or partial similarity to other sections of itself. Referring back to the foam bubbles, notice how there exist some bigger bubbles with smaller ones in between them.

If one were to zoom in further on these small bubbles, you would see the small bubbles have even smaller bubbles between them. All of a sudden what I previously referred to as small bubbles, become the big ones, and the even smaller ones have simply replaced the initial small ones. You can see this effect in the pictures. The big bubbles can clearly be seen, and a zoomed in section shows the smaller bubbles. While these bubbles are aesthetically pleasing, there are more intricate fractal patterns which look like something one might present in an art gallery. These fractals are created when looking deeper into certain mathematical aspects and sequences, such as the squaring and adding of complex numbers.

The Mandelbrot Set

This squaring and adding of complex numbers leads to a very fascinating fractal pattern known as the Mandelbrot set. This sequence for creating this can be summed up through the following equation:

To sum it up in words, one starts with an initial complex number, c, squares it, and adds it to itself.  Following this, the new number is taken, squared and added to c, the original number, again. A very neat and tidy equation which creates the fractal on the left.

This series should diverge, after all there will be an infinite amount of points. While it has a definite boundary, this boundary has an infinite length. With this concept of visualizing infinity it has been named the “Thumbprint of God”. This fractal showcases the self similar nature which was previously discussed. You can see smaller black circles stemming from the main one, all of which, when zoomed in will show the original design. In theory, you could zoom in forever and be seeing the same patterns. The Mandelbrot set above is a simple one generated through some short python coding but other more detailed depictions show the finer intricacies of this fractal. To the right is an example of the Julia set, a fractal related to the Mandelbrot set and below is its formation depending on the scale of the perturbation used to generate it. Perturbation is just a fancy way of saying it gets a little nudge.

Hofstadter’s Butterfly

Diving deeper into some of the more advanced topics of physics, we come across the fractal known as Hofstadter’s butterfly when studying Bloch electrons.

The fractal’s name comes from the interior of the graph, the gaps in between the main lines, which resembles a butterfly. The graph and the research tied to it holds many links to the quantum Hall effect, however this is a topic which would require its own section to go into. The fractal pattern was replicated in experiments since 1976, the year when Douglas Hofstadter first made the discovery.

Koch Snowflake

Moving on to another well known fractal, and a very easily made example at that, is the Koch curve, in particular a snowflake design. The Koch curve, named after Helge von Koch, follows a simple rule in that each straight surface gains a triangle, which then leads to 4 straight lines from which new triangles are born. The following picture shows exactly how this works and the resulting snowflake pattern we get after several iterations.

Each new triangle leads to an increase in the length of the perimeter of the surface. After an infinite amount of iterations, it leads to an infinite perimeter length. This concept of an infinite perimeter can be applied to other things such as coastlines, maps and rivers. Many things which have some sort of boundary gain small increases depending on the detail through which we see them. If you zoom into a map, you’ll start to see straight lines when in actual fact there should be some form of curvature. Just like the Koch curve, each extra detail added, adds a bit to the length, and if one were to zoom down to the tiniest details, you would be stuck going to infinitely small details.

Conclusion

The fractals I have discussed above only scratch the surface of the hundreds of fractals that exist around us. Many of these fractals are fascinating and have complex patterns, and in all cases one can zoom in to an infinitely small level and still recover similar patterns to the ones you see at first. Some of these breathtaking fractals link into more complex topics in the world of physics, others are simply fascinating to look at. And some as we’ve seen, challenge our view on things which we previously thought had fixed length.

References

1. M. Field, “Mathematics through art–art through mathematics,” Proc. MOSAIC, pp. 137–146, 2000.
2.  D. R. Hofstadter, “Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B, vol. 14, pp. 2239–2249, Sep 1976.